Taking the extremal limit of a non-extremal Reissner-Nordstrom black hole (by externally varying the mass or charge), the region between the inner and outer event horizons experiences an interesting fate--while this region is absent in the extremal case, it does not disappear in the extremal limit but rather approaches a patch of \(AdS_2 \times S^2\). In other words, the approach to extremality is not continuous, as the non-extremal Reissner-Nordstrom solution splits into two spacetimes at extremality: an extremal black hole and a disconnected \(AdS\) space. We suggest that the unusual nature of this limit may help in understanding the entropy of extremal black holes.